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A rational model foe bod-exertion

IN STREAMS

A Thesis Submitted

In Partial Fulfilment of the Requirements
For the Degree of

MASTER OF TECHNOLOGY

/ 00

A.R. RAMANI

-n

I. I

CENX :;; Ah J~$B74 A.RY s

to. JVe. I I

C£ _ , 94S- n- RAM- R.

JUNE‘76

\ /W^

£27 j 2
E 1-1 i n.

t# the

Indian Institute of Technology, Kanpur

July, 1968

CELTIPlCATiS

This is to certify that the present
work has been done under my supervision and
the work has not been submitted elsewhere
for a degree or a diploma#

Dr* Guru Bass Agrawal
Associate Professor
Department of Civil Sngg,
Indian Institute of Technology
iianpur.

table op contents

Page

Abstract i

Acknowledgement 11

List of Tables ill

List of Figures iv

Chapter

I. INTRODUCTION 1*7

a. General

b. Scope and Objectives of Study

II/ , REVIEW OP LITERATURE 8 - IS

111,1 THEORETICAL COM SID© ATI OHS 16-28

a# Basic of 100 Determinations

b. Order and Molecularity of Reactions
c* Kinetics ©f SOD Reactions ;

1, General Observation
2* Miehaells-Menten Hypothesis
Applied to BOD Reactions
■: Model; ,/

1, Mathematical Formulations

2*. : Mghlfioaaoe'' : ’®f '

. 3* As sumptions la the model

iir© //.--ib^npKEa^^ ■

' a. Experimental Technique fer BOD Betermina*
tion

b. Measurement of Bacterial Growth \

c. Substrates used In the Experiments ,

&*. Seeding Conditions
e* Sets of Experiments

?• mmm® 3 2 -m

is. jusmsxa mb ahalysis m mmus 4M2

a* Theoretical Curves for BOT) 'Progression
fe. Theoretical Carves for Bacterial Growth
c* Agreement of the Observed Data with, the
Rational Model,

d* Comparison with the Streeter- ehelps Formulation
®* Q® termination of the Constants that appear in
the Model '

f. Practical Applications
¥12. COZICLUSlOH

■ Recommendation for Additional Research
1IST OF REFER IICRS
GLOSSARY OF SYMBOLS
APFUIB2C&S

63-04

.6ft::

abstract

The inadequacy of the existing formulations describing
the BOD exertion in streams is brought out clearly and atten-
tion is focused to the major significant role played by bacteria
in the removal of BOD. A mathematical model incorporating the
salient features of the BOD-reactions is suggested to describe
the profiles of organic pollution in streams, it is founded
on the Mi chael i s- Me nt en kinetics of enzyme- catalysed reactions.
This rational formulation together with the Monod*s equation for
bacterial growth go to predict the pollution in streams to a
fairly accurate degree. The solutions of these equations are
worked out by digital computer techniques. An experimental
investigation is presented for the verification of the model
and the results indicate a striking agreement between the two.
The application of the rational model is stressed for carrying
out river- pollution abatement programmes based on more such rea-
listic and rational approaches than on the existing crude formu-
lations, •

***• author uishea to «kj press his sineer®
gratitude to Br. Ouru Data kgrmml ehose guidance
aM help enabled the studies to he undertaken and
to eoeiplete sueeesafully*

Profound thanks are herein expressed to
Br» Kashid 1, Siddiqi and hr. A.V.S, Prabhakar Hao
for their useful advise frost time to tine*

Ltm OF TABLES

TABLE

S®ts of Experiments

Significance of Seed, to Substrate Latio

Values of 8^ from' observed data

Values of K g and ^ a3g fro© observed data

LIST <F FIGURES

Flgmr# Pag§

3U Observations for Sot (l) 33

(a) BOG Progression (b) Bacterial Growth
t* Observations for Set (2) 34

(a) BOD Progression (b) Bacterial Growth
* 3 * Observations for Set ( 3 ) 35

bod Progression (b) Bacterial Growth ■

4* Observations for Set (4) 36

(a) BOD Progression (b) Bacterial Growth
6. Observations for Set (5) 37

(a) BOO Progression (b) Bacterial Growth
6» Observations for Set (6) 38

(a) BOB. degression ■ (b) Bacterial Growth :

7* Observations for Set (?) 39

<*) BOB Progression (b) Bacterial Growth
S* Theoretical Carves for BOD Progression 41

/ . ;Bff#«t of

Theoretical Carves for BOD Progression ^ 42

Effect of Variation in

.10* Theoretical Starves for BOD Progression 43

Effect of Variation in It

»aac

11* Theoretical Garves for- BOD Progression 44

Effect of Variation in X^

3J* : -theoretic^ for BOD Progression : ; ■' : 45

Effect of Variation' in S_

13* Theoretical Correa for Bacterial Growth 47

(a) Effect of flirtation in K

(h) Effect of Variation la g_

14* Theoretical Corves for Bacterial Growth
(a) Effect of Variation, ia
(h) Effect of Variation ia S

14* Agreement of the Observed Bata with the Theoretical Bata* 52

16* Goffiparisoa of Streeter- Phelps and national Model (Glucose) 54

17* Comparison of Streeter- Phelps and national Model (Peptone) 56

18* Comparison of Streeter- Phelps and Rational Model (Sewage) 66

10* Determination of Constants t 8L # Xg* 8^^ 60

20* Calibration (Serve for Bacterial Cell Concentration 70

chapter I

IHTROOCrmOJ?

a* General *

For the maintenance of normally satisfactory conditions
in a river, the oxygen economy is of paramount consideration*

When pollutional nuisance of receiving waters is to he avoided,
the DO and BOD, taken together, are generally relied upon to
delineate the profile of pollution and natural purification
on which engineering calculations of permissible loadings are
based* Requisite evidence is rapidly accumulating on the in-
adequacy of the existing mathematical formulations to account
for, in a more realistic manner, different variables that go
to describe the nature and extent of pollution and the microbial
population* In particular, the scope of the classical aspects
of BOD-klnetlcs needs broadening to an extent that It will engulf
a good deal of the already existing knowledge from the related
domains like biochemistry, Chemical-kinetics etc* The obvious
rational approach will be to lay stress on the formulations of
mathematical ' models based on a deeper understanding ’ of the princi-
ples Involved* The inherent complexity of such formulations, as
it may, at first sight seem to be their major disadvantage for
practical applications, can be easily overcome with the use of
modern digital computer techniques* The author’s work -follows
the path of its predecessors in attempting to set up a realistic

and rational theory for the progression of BOD In streams .

The classical kinetic theory In this area stems primarily
from the early works of Streeter and Phelps , 1 to who® the honour
of its establishment is due. The first-order, monomoleaular
equation, they suggested for the graphic path of the carbanace-
ous BOD curve, stipulated that the rate of biochemical oxidation
of the organic matter is proportional to the remaining concentra-
tion of the unoxldised substance. Their formulation was an
enormous oversimplification of the reactions taking place in an
environment characterised by very complex parameters - that were
physical, chemical and biological in nature* Speculations were
mainly raised against the extreme flexibility of the first-order
equation which seemed to produce almost any conceivable type of
curve with indefinite values of the constants* Small differences
In the shape of the experimentally observed curve could produce
great fluctuation in the computed values of the constants, so
that little physical significance can be really attributed to
them. The theoretical basis for the application of such an
equation to biochemical oxidation was derived from the fact that
many simple chemical diffusion and reaction phenomena follow
the monomole cular pattern. However , no due recognition was
paid to the fact that the biological oxidation of sewage and
other pollutional organic matter Involves a complex microbial
flora which' acccmi^sh ;:the\exidati<^-l^f ' thousands of inter-
related enzyme reactions oeeuring simultaneously during cell .■ ,
metabolism and multiplication. Thus the overidll

bacterial respiration are indeed ©ore complex than is often
realised* It is evident that the Important facets of the
BOD-kinetics are inexplicable by the first-order equation*

In short, Streeter- Phelps formulation will only be a crude
simplification for the ease of streams receiving wastes from
communities and industries*

Since the classical work of Streeter and Phelps* several
modifications were suggested by investigators like Thomas 8 *

Fair 4 , Oxford and Ingram 2 etc* Evidence has accumulated In
recent years on the discordance between the experimentally
observed curves and the theoretically established ones by the
above-mentioned research workers* The most notable reason for
this lack of correspondence can be easily tradeable to the absence
of a term in these equations representing the bacterial concen-
tration* After all* it is the faculty of bacteria that is res-
ponsible for the removal of BOD and it seems unjustifiable to
have denied its due importance in the BOB-equatlons. Actually,

' this has been acknowledged by Phelps himself *

$ m

Recent research works pursued by Busch and Gaudy et al'
have demonstrated the presence of more than one phase in the
carbonaceous BOD curve* The diphasic curves they observed for
different substrates under various seed conditions exhibited a
"plateau** that separated the rapid phase of oxygen uptake from
the’ slow, .phase*: , .On the basis of their experiments! a theory ;
; vas’ : propounded which ascribed ' tbeV|rogvosll^ .variation in BOB
values to the effect of varying ratios of baeterla to higher

organisms in the seed population. Having established that the
progression of BOD in soluble substrates vas a two- stag© and
not a first-order reaction, they concentrated their efforts on
the re pr educability of the plateau and its subsequent relia-
bility to suggest a short-term BOD test. However, some uncer-
tain! ty still prevails regarding the existence of such a plateau

since Butter field ' s observations have expunged such an exis-
tence, Other important developments include the mathematical
two-phase formulations of Garret and Sawyer 13 for bacterial
growth.

The exact interpretation of the BOD kinetics confronts
the engineers with many problems that have blanketed this area
to a great extent in the past, (The problem is as analogous
to that confronting mice who, in Aesop's fable, wondered how
to hang a bell around the cat's neck to warn them of its pre-
sence), It is extremely a complicated task to propose any
mathematical model, based on the kinetics of Individual bio-
chemical reactions involved in the exertion of BOD In the
absence / of a refined analysis, 'it becomes necessary to view
the ' bacterial klnetl cs ' with a ; 'maeroapproach f : to gain atleaet ,
some insight.

The proposed model tor BC® pr^ author,

invokes the universally established Mlchaelis-Menten hypothesis
for the rational Justification of its formulation. It is well
'founded' bacterial growths in ehemi-

stats. The revelation of his works focussed considerable

attention of the research workers towards the deeper understanding
of the principles involved in the progression of BOD* Recent
experiments on the study of concentration effects in the hiologi-
„ cal oxidation of trade wastes by Wilson 16 have corroborated the
conclusions derived by Monod* Wilson sought to extend the know*
ledge of basic principles towards the achievement of low* cost
treatment systems operating at optimum biochemical efficiency*

Parallel to the Mi chaeli s-Menten type of enzyme* catalysed
reactions, the BOD equation can be written as,

Bacteria * BOD ~,.. 03 9 rgeri Increased number

(Substrate) of bacteria

♦

Oxidised products
(00 2l BgO, KH 3 etc.)

Applying the laws of mass-action to the above equation, it is
evident that the rate of BOD exertion will be proportional to
the concentration of two substances vis,, the engyme (that is
contained in the bacterial vessels) and the substrate* Even
though a closer investigation of the metabolic activities of
bacteria may reveal the involvement of a large array of ensymes,
as a simplification, the concentration of bacterial- cell-mass
: : can generally ■ be: , consii croft. ■ t * represent the ; total . catalytic y ■ V
aetivttyyia the' syetsm* .h ;

; fibs mathematical model presented by the author for the
progression of BOD in streams emphasises the presence of a term
in the BOD- equation to take care of the bacterial population*

It incorporates as many constants as there are variables and

it is suggested that a compilation of BOD- curves for different
conditions be prepared after solution of the proposed equations
by digital- computer-techniques. Once this is done, the applica-
tion of the derived formulations becomes facilltatlve for routine
works. It only remains to aim for the proper match of the experi-
mentally observed BOD- curves with the theoretical graphs.

To sum up. The model is rational because it recognizes
the due significance of the role played by bacteria in the biolo-
gical purification of wastes, dumped to the streams j It has got
sound bearing in the annals of enzyme - chemistry of catalysed
reactions; It describes adequately the full gamut of BOD-klnetiea;
Its application is more direct and facilltatlve than that of a
first order equation. The author has undertaken experiments to
validate the theory proposed* It is sincerely hoped that due con-
sideration will be given towards the application of such realis-
tic and rational formulations by those who want to lean on more
scientific approaches than on the existing crude formulations
while pursuing river- pollution- abatement programmes.

b, Scope and Object of Study :

As already indicated in the previous section, presentation
of such a model as to adequately describe the BOD progression in
streams based on a deeper understanding of the bacterial kinetics
involved, has been the main aim of the author* Efforts have also
been taken for a comparison of the rational model with the Streeter-
Phelps-equation, primarily to bring out the inadequacy of the later
formulation in dealing with the practical situations*

The study, in short, consisted of the following two parts:

1, To formulate a rational model for the BOD progression in
streams and demonstrate its use in the practical problems
of r i ver-pollut i on- e v&lu&t 1 on •

2* To undertake experimental studies to look for agreement with
the proposed theory*

The numerical solutions of the mathematical formulations
are to be worked out with the help of digital computer and th*lr
graphic paths, traced. It is easy to compile the different sets
of curves for different values of the constants, under various
■ initial ' conditions. , ; fhe : .; author'^

as the work will otherwise become quite voluminous* He prefers
to demonstrate the use of some ’model* graphs and he would like
to pinpoint the necessity of preparing the various sets of BOD-
graphs by the concerned agencies in the area of river-water-
pollution- control *

CHAPTER II

REVIEW OP LITERATURE

The major findings of the early works on BOD kinetics,
can b®, for better understanding, categorised in the following
main three groups*

1. Those, which were reported as modifications to the Streeter-
Phelps - first order equation that suffered very much from its
extreme flexibility in the determination of the constants. The
investigators of this group attempted to strike at a close
correspondence of the experimental data with the theoretical ones.

Of notable mention are the "lag formula” developed by Thomas 3 ! the
"retardant" BOD equation formulated by Pair and the "logarithmic

BOD equation" suggested by Oxford and Ingram. Their works revealed
only one phase in the carbanaceous BOD curve.

2, Those of very recent times, that describe a new type of
kinetics for the course of ^sipehsaaeeed* BOD comprising more than

' g *7

one phase. Research works of Busch and Gaudy et. al are of
notable mention in this connection. They investigated the influ-
ence of the higher form of microorganisms on BOB progression ©f
soluble substrates and reported the presen®* * "plateau" or
discernible pause separating the earlier rapid phase of oxygen 5
uptake from the second stage. Based on the diphasic formulation,
Busch suggested a short-term BOD test (T fc oD) that involved tracing
the BOD curve to the plateau and determining the cell oxygen equi-
valent at that point. It should also be noted however that

Incidentally the experiments of the aether too} have shoved
no evidence of the occurrence of such plateau in the earbanaeeous
BOD curve. Thus there lie# still a great degree of disere*
preaparing or atleast inconsistency regarding the occurrence and
implications of the plateau.

3* Those , which interpret the kinetics in a mathematical fashion
and are of particular significance for practical considerations.

They are wall Bounded on the theories already established (or) at*
least on the observations , frequently noticed, of the two wain
methods of representing the kinetics of the biochemical reactions .
involved in the BOD progression, one was due to Monod, which was
subsequently used by Einshel wood Herbert 3,1 , Downing* 2 and
others. It relates tile growth rate of the active organisms (which
is a measure of rate of purification) by the well known Michael! s-

13 14

by Garret and Sawyer and adopted by Eekeaf elder and McCabe

postulates two growth phases * one in which the substrate concentra-
tion Is always greater than a threshold value required for the
complete saturation of the afttlve enzyme centers of the organisms,
the other in which the degree of unaaturatlon increases as the
■substrate le progressively utilised as food* These two postulates

still require Intensive data collection for their unwarranted reeogf-
nition in their adoptability for the highly engineered verbs in

the field of biological systems* Driven by lack of confidence

in the existing kinetic concepts, I.S* Wilson 16 undertook

special experimental studies the result of which he interpreted

in the light of both the Monod and the two- phase theories. His

published data fitted well the Monod *s theory, Clarks et al

and Charles S. Revelle were fas elated towards second order

BOD equations. Very recently, a stochastic model for BOD and

DO in streams has been presented by Richard P. Thayer et al.

Since the author’s work comes under group - 3, mention
should be made in brief the methods of approach of the various
Investigators of that group in particular, towards the y rati-
onal considerations of the kinetics involved.

In 1942, Monod conducted a quite complex study of the
growth of pure cultures of E. Coli and B, Subtllis on an assort-
ment of single carbohydrate substrates. He was able to fit his
data very closely to the rate concentration curve of the same
form as the Michaelis-Menten equation for the rate of en z ymatic
reactions, viz. ,

where ,

growth rate of cells

maximum growth rate when the

substrate Is unlimited.

Kg * substrate concentration at which
the growth rate observed is one
half of the maximum value | satura-
■ tion constant*

S * Concentration of the substrate

Verifications of Monod's conclusions were done by
19

Mr. Andrew L. Gram in 1956. After conducting extensive

experiments, Gram found that the rate of removal of substrate

was closely described by Monod equation. He, as well as Stewart
20

and Ludwig noted that the removal rate per unit weight of
organisms was a function of substrate concent Bati on only if a
constant conversion of substrate to organisms was assumed.

Earlier, in 1952, Garret and Sawyer postulated that
there were only two phases- a log phase and a transition to the
stationary growth phase - that were of "practical” importance
in defining the reaction kinetics of aerobic biological processes.
Even though they agreed that Monod 1 s equation, was well founded
theoretically, they declared that the observed data were in. better
agreement with the two- phase formulation than with Monod's equa-
tion. Their main objection to Monod's relationship was that "the
equation denies the existence of a constant rate of growth above
critical concentrations of food, although this is the most fre-
quently observed phenomenon related to the growth of bacteria***

$ he two phase formulation for bacterial growth was comprised of
the following two equations:

In Phase I, ■

& *

dt f

In Phase II,

H -

where , X * Bacterial concentration.

S * Substrate concentration
a « Increase in organism concen-
tration produced by unit
decrease in substrate con-
centration*

^ and Kg * the rate constants.

The results Of Garret and Sawyer also showed that the kinetics of
oxygen utilisation by mixed organisms were similar to those for
pure cultures, although the rates were lower*

Sometime in 1962, 1, 3. Wilson*® undertook extensive lab
studies to seek refinement of the theories of Monod and Garret
and Sawyer* He tried to evaluate the constants in their equations
with notable success. His main concern was in the practical utility
of these equations towards the development of high-efficiency , low-

ft*.

cost biological-waste- treatment-systems. He emphasised the need:'::
for rigorous control of experimental conditions, for even alight
errors in fixing one or two points in the rate curve would cause
large error in the interpretation of the results. His published
data fitted better the following Monod-Type-equation than the two

phase formulation.

where

* the reciprocal of “yield co-

efficient” which is the increase!

in the cell concentration per
unit substrate-depletion.

(KL.. K , X and S denote the same
A max

quantities as defined earlier*)

The major limitation of his paper, as pointed out by Mr. P.H.

McGauhey # was that he did not make any conclusions particularly
evident.

In 1964, Kesha van at al conducted studies on the kinetics
of removal of organic wastes, in connection with their proposed
rational formulations. The correlation between the experimental
data and the theory was very good. They invoked the well estab-
lished Mi chaeli s-Ment en hypothesis for the justification for their
rational formulations,

The second-order BOD equation suggested by Clark et al
in April 1965, was of the following form?

" H 8 KS 2 where,

K Is the rate constant

This was based on the following reaction,

2 8 — ■> Products*

i,*e» ' ■ 2 (10D) — > Products (like BgO, NHg etc*) :

With their;, experimental evidence they concluded that n a second- ;
order BOD equation is as good a fit as the first-order equation
for the observed data” and that "a second-order equation ©an be
solved with more facility than ©an an equation based on a first-
order reaction.’* Challenging the rationality of their model,

'' ' j||t '■ - ■

Keshavan et al severely objected that the model could not be

justified even if the Michaelis - Menten hypothesis was invoked
by the authors* Their main argument was that the rate of reaction
should be proportional to the concentration of two substances ,
namely the substrata and the enzyme, Clark et , al , they concluded,
were not at all justified to have proposed the above equation*

The works of Kesha van et al mentioned earlier, contributed
significantly towards the development of second order Bio- oxidation
kinetics by Charles 0, Rove lie et al 17 in December 1965, The
latter suggested that the rate of BOD removal was proportional not
only to the concentration of BOD remaining, but also to the con-
centration of bacteria, each raised to some power, as follows i

Rate of BOD
removal

Concentration
BOD remaining

Concentration
of Bacteria

Since the exact determination of the values of n and m in the above
equation remains still a puzzle with the present knowledge of
enzyme kinetics concerning BOD reactions, the value of n and m was
taken as 1 in order that the equation may be amenable for further
mathematical /treatment, ; In the period before ; the ; -flieet::
ous respiration, their second order equation provided a realistic
and rational approach to the kinetics of biological oxidation, '

The stochastic model for BOD in steams presented recently

by Richard F, Thayer et al , assumes, however, a first-order
reaction-rate for the or gani ®- pollution-removal*

The literature study brings out dearly the necessity of
isolation of an area where added research is highly needed before
certain current postulates can be used with confidence. The area

not just the formulations, hut their #ldely recognised acceptance
by many for their useful applications. This calls for a deeper
understanding of the basic principles involved. The difficulties
that surmount the accurate prediction of kinetics in this area
need not be cataloged as they are the ones, very often, encountered
in association with any biochemical reaction Involving thousands
of enzymatic systems at intr a- cellular levels. Thus the Inherent ■■
complexity that has blanketed the area is to a great extent
unavoidable unless there is a major break through by investi-
gators who will propose methods for the exact determination of
the order and molecularity of the complicated, deeply Involved
biochemical reactions. However it is also possible to gain
. sufficient;, insight:,.. in;the .light 'of the'; exi^^
the f GBTOlation of rational expressions on basic

theories and adequately supported by practical observations .

CHAPTER III

THEORETI CAL CONSI DERATI OKS

a. Basis of BOD Determinations:

The ultimate foundations underlying the biochemical oxygen
demand of waste materials are the enzyme catalysed processes
involved in the growth and multiplication of organisms acting on
these materials. Oxidation, as it applies to biological and
chemical processes, may be described as the loss of electrons
from a substrate. The primary biological function of Oxygen in
aerobic systems is that of a terminal acceptor for electrons
liberated by oxidation reactions carried out during cell meta-
bolism. The electrons are sequentially transfered along a res-
piratory or ‘electron transport* chain through the coupled cyclic
action of several carrier systems to oxygen, the terminal accep-
tor, yielding water as an end product. The ‘cytochrome chain*
includes the substrate, HAD*, flavoproteins, several cytochromes
and the terminal acceptor, oxygen. Hence it is reasonable to
expect that the uptake of oxygen during cell growth should be in
direct proportion to the sum of all the oxidative processes occur-
ing within the cell. As it is known that the extent of cell growth
is directly proportional to the quantity of biologically oxidi sa-
ble substrate present, it is possible to indirectly assay the
‘strength* of the waste materials by measuring the quantity of
oxygen uptake by the system. This is the basis of BOD determiaa-

tions. Further it is well understood that the rate of oxyg© n
uptake is a function of the rate of substrate oxidation by ‘t* 1 ®
cell, and hence, ultimately traceable to the rate of enzyme
reaction.

b. Order and Moleeularity of Reactions:

Before considering the kinetics of BOD reactions, th®
salient features of ordinary or 'uncatalysed ' reactions shall
be reviewed with specific reference to their order and molecu-
larity.

In accordance to the law of Mass Action, ’the rate of
reaction at each instant is proportional to the concentration
of reactions; e a ch concentration being raised to a power equal
to the number of molecules of that reactant participating in the
process', 'Kinetic order' refers to the sum of the exponents
of the concentrations which determine the reaction rate. 'Re-
action Velocity' refers to the time rat® of change concentra-
tion of the compound considered. This denotes generally the
rate of reactant disappearance, whilst, in enzyme reactions of
biochemistry, it applies to the rate of product formation. It
is to be pointed out that for reactions with no intermediate

products, the rate of reactant disappearance is the same as th®

■ ■ %

rate of product formation, where the stoichiometric ratio of
reactant to product is unity.

In general for a reaction involving r t , , . *

molecules of compounds A, B, . . , , the rate equation

given as follows:

i J£

qA + rB + . . • — - " '■' > products

v = - da/dt ■ * K a q b r . . . . .

where a & b . . . are the respective
concentration of A, B, , , ,

The reaction order, n * q + r + . . • .

Here the n - molecules must interact with each other in the
rate-limiting step of a reaction, and the rate equation for
this n-th order reaction contains n - concentration terms,
and the rate constant has the dimensions of concentration^ 1 "* 11 ^ time"" 1 .
It should be however borne in mind that, it is statistically highly
improbable that all the n - molecules participate simultaneously
in a rate- limiting step. They may proceed, by a series of any
reaction order. For example, a termoleeular reaction may proceed
by a series of bimolecular, and/or monomolecular steps. Reactions
of this type are klnetically of first, second, thiJd or mixed over-
all - depending on which steps are rate- limiting, and even fractional
orders are not uncommon.

For example, the reaction 3 KC10 ? . — * KClO^ +. 2 KC1
will be of 3rd order, if molecularity were to equal order. But,
actually this is a second order reaction in KC10.

From the foregoing, it is evident that the determination of
the order of a reaction is the most important part in describing
Its kinetics and that order of a reaction need not equal molecu-
larity. In the following discussions, considerations will be given

1 $

to the theoretical order and molecularlty of BOD reaction.

c. Kinetics of BOD Reactions:

1. General Observation:

The BOD reaction can he described^' as,

Organic Matter

(BOD)

Oxidised products
(H 2 0, C0 2 , NH 3 ) etc.

For illustrative purposes, let an organic compound, glucose be
considered. Oxidation of glucose yields COg and water. This
can be represented as:

= 6=12 °6

bacteria

6 COg + 6 HgO

But the actual sequence leading to the formation of products
is far from simple. Phosphoenol pyruvate which is formed via
the glucolysis path way, enters the Krebs cycle in the form of
Acetyl CoA. The Krebs cycle is as complex as the proceeding
steps. Thus the equation 1 is just a summation of the thousands

. r .

of inter-related enzyme reactions occuring simultaneously during

cell metabolism and multiplication. It becomes clear that the

over-all kinetics of bacterial respiration are more complex

than is often realised. A more extensive discussion of enzyme

reaction mechanisms may be found in the work of Reiner .

From the previous discussions, it is quite evident that
the oxidation of glucose by the bacterial cells can not be a
first order reaction, because of the number' of intermediates

between glucose and the final step. Each intermediate step has
its own or d ere# of reaction and Involve different enzymes.

Further the total enzyme concentrations of the system increase
with the replication of the bacterial cells. Hence one must
admit the lack of knowledge of molecular ity, and inability to
test the order of BOD reactions, before proposing any rational
model. When such is the case for only one substaate like glucose,
it is not difficult to realise the degree of enormous complexity
involved in predicting the exact kinetics of the biochemical
oxidation of other heterogeneous compounds,

2, Michaelis - Menten Hypothesis Applied to BOD Reactions:

With the dearth of knowledge already admitted, regarding
the order and molecularity of the BOD reactions, one has to take
a 'macroscopic' approach to describe the BOD kineties. Since the
BOD • reaction is entirely biochemical, the Michaelis-Menten
hypothesis can be invoked for a formulation of a similar expre-
ssion. An enzyme - catalysed reaction of Michaelis-Menten type
can be represented as follows:

' V / _ % . _ K 3 _ _ _

E + 3 A. e S + P + E fe)

Where E is the enzyme, S represents the substrate, E S
denotes the enzyme - substrate complex and p stands for the
produet. K^, Kg, and arb the rate constants in this
reaction.

If the value of Kg is far less than that of K^, i*e,
if the rate of product formation is controlled by the specific

rate Kg in the sequence of reactions expressed by equation (2) ,
the following can be written,

L ES

v = (3)

Kjj + S

where v is the rate of reaction,

Kjj - the Michaelis - Menten Constant,

E - the enzyme concentration

From (3) it is clear that the rate is reaction is thus
dependent on the concentration of two substances, namely vie
the enzyme and the substrate.

In the same fashion, considering the concentration of
the bacterial to represent the enzyme concentration (at least
in a 'macroscopic* way), the macro-reaction for BOD would
be written as :

°^lub 3 trate) 6r * Bacteria -23 3 5“ More + Ba«teria

Oxidised pro-
ducts .

r Im light of the 'previous discussions, the following
second-order blmolecular reaction would be written for BOD
exertion,

- -4- (BOD) * K (BOD) (Bacteria) ,
at

each quantity within the bnackets
representing respective concentration.

It was this form of BOD expression, which Charles S. Revelle
et, al proposed and their experimental results revealed a
good striking coincidence with the stipulated model.

d. The Rational Model:

1. Mathematical Formulations:

The rational model has been based mainly on Monod*s results
(on bacterial growth in a chemostat ) , that behaved in a manner
typified by Michaelis-Menten equation. The expression he suggested
for the bacterial growth was of the following form:

w s

K s + S

Where p.

- growth rate of alls

(i.e.

1 $£
x dt

K s - substrate concentration at
which the growth rate observed is
one half the max value; saturation
constant .

X - Bacterial Concentration at

time : ; v ':;t

S - Substrate concentration at

Equation (4) can be written as:

or,

dx *Wx x s ^

dt “ (K g + S)

Assuming that an increase in the bacterial mass is taken
as directly proportional to the amount of BOD removed, it can
be written as,

(X - X Q > * <S 0 - S) (6)

Where, KL is the reciprocal of the

well known 'yield coefficient'

X ■ Initial bacterial eoncen-
o

tration,

S_ = Initial substrate concen-

■

tration.

Differentiation#? of (6) with respect to the third variable, time,
denoted by 't', leads to, /

M * . id -1

dt dt %

and substitution for dx/dt from equation (5) in equation (7)
results in the following equation, describing the kinetics of
BOD progression.

% %ax x |
(C + S)

Equation (8) is a^s© the pise fern of the equally adopted by
Wilson.

Equation (f) is a second-ord^r differential equation in ®
It contains 3 variables, via. S, X and t. Substitution for %
from equation (6) in eqn, (8), permits integration of the later
equation in only tv© variables 3 and t, as fellows*

~ * K * >w

&g * s

( x ® ■* Hejj * s °

i f ©I

dS (Kg * S)

(X* + — .1

Ivj£

Integrating both sides, within appropriate limits,

* dS (Kg e 3)

(x 0 * (S 0 ~S))

® ,% : ^S3

■ Equation ■ (9) takes the final for® after ♦integration
as follows*

llvv %

e% '%»**: *

^ *“” 4 "” ( 8 a **85

x o

is show® in the appendix.

Equation (10) is the proposed kinetic expression for the
substrate depletion in streams. It will be recalled that the cons-
tants K ma3c and Kg g® to describe adequately the rate behaviour of
the BOB reactions with the initial conditions given by S 0 and

Equation (5) describing the bacterial growth can be invoked
again for further mathematical treatment* Substituting in this
equation from equation (6) , the following differ-

ential equation is obtained.

* ^max % ^ )

(Kg+ ^.Kj (X- X*) )

( 11 )

Rearranging equation (11) and ^integrating it between the limits

t, X

X ) the following equation

showing the relationship of bacterial concentration with time,

X « X,

r 3,,-Kj a 0 -x^

toms" (10) and (;

w t mi

x «

%*% x c

which characterise the kinetic pattern

■<C. BOO progression and bacterial growth respectively! are amenable to
solution by digital computer , thus a compilation of different sets
of graphs, for various values of the constants under given initial
conditions can be prepared * The experimentally observed graphs for
BOB progression can be compared with the theoretical ones , and the
predictiomof river-pollution is made possible to a remarkably degree
with the knowledge of the nongtaats, .JL* . B^ ax and K^.

2* Significance of the Constants:

Mach of the practical significance of the proposed mathe-
matical formulation is gained from the presence of the three constants
viz. i K x (the reciprocal of “yield coefficient") , K g (the saturation
constant) and K ffiax (the max-growth rate constant) * These three constan-
ts, along with the initial conditions X 0 and describe the organic-
pollution - profiles in a stream in a more realistic manner than the
constants of the first-order reaction do*

is the reciprocal of “yield coefficient." Thus 1/K^ refers
to the amount of bacterial-mass that has been assimilated in the
depletion of unit amount of substrate. The assumption of the linear
relationship between the bacterial mass and the amount of BOD-removed

? g gg 0*7 go

* * * * %

as one will expect, remains constant for a given substrate and organism,
and is dimensionless.

and Kg are the two constants, standing far the kinetic
properties of bacterial reactions* is the maximum value of

growth rate at infinite substrate coneehtration (i.e, S ) and
khus has the dimensions of inverse time. It has thus a well defined
meaning both operationally and physically.

The saturation constant, K g , can be stated mathematically as
follows:

i.e. It equals the concentration of the substrate, S at which the

observed growth rate is half the maximum growth rate. Both the

constants and vary with the organism and substrate but do

not seem to be dependent on temperature to a significant degree at

maximum growth rates.

3. Assumptions in the Models

In view of the fact that the difficulties towards the
development of a comprehensive kinetic theory for bacterial oxidation
can not be easily surmounted with the existing knowledge in that
area, the proposed model by the author leans heavily on the following
assumptions that are compulsory.

A principal assumption i3 that the total catalytic activity
of the system is represented by the bacterial concentration. When
a bacterial cell multiplies, all of is its constituents are assumed
to be identically reproduced and hence the enzjmic concentrations
increase in direct proportion to the increase in bacterial mass.

The kinetics of mixed culture is considered as though the
mixed culture were pure. This assumption has been already validated

.TU9

by Garret and Sawyer.

Additionally, an increase in the bacterial mass is taken as
directly proportional to the amount of BO© removed* This has been ,
verified earlier by many research workers*

Finally, oxygen concentration is not a limiting factor, i.e.
thebe is enough supply of oxygen. But for this assumption, the
mathematical treatment of the model would have become much involved.

It was also assumed that the variables like temperature, pH eto.

are maintained at an optimum level in the environment and that the

supporting medium contains all the essential growth factors and
no inhibiting substances.

CHAPTER" IV
EXPERIMENTAL METHODS

a. Experimental Techniques for BOD Determination:

The experimental data for BOD were obtained with the Warburg
Techniques rather than with the standard dilution techniques, A
number of factors were responsible for this choice. The standard
dilution test requires a large number of BOD bottles because of
short time intervals (4 hours) , whereas only one manometer and
flask will be needed for each dilution of the seed in the Warburg
method, Also the BOD curve can be followed in the early critical
stages very easily and accurately by the manometrle techniques.

The shaking of the reaction flasks was kept minimum, s© that
this will , to some degree, correspond to the normal stirring in
streams,

b. Measurement of Bacterial Growth:

For measuring the bacterial growth, optical density which
measures In turn the bacterial density was made use of in the
experiments. Bacterial density is defined as the dry weight of
bacterlaper unit volume ©f the solution. It is more closely
related tOr the quantity of the bacterial protoplasm and henee
to the enzyme activity. Optical densities for this study were
measured on B & L Speetronle - 20 at an optimum wave length' of
$90 m u* the calibration curve for bacterial concentration is
shown in Appendix A. Increasing concentration of the bacterial

. 30 ,

calls for calibration purposes were obtained by centrifuging
the domestic seed at^g,

c. Substrates Used in the Experiments s

. . T he: .substrates ' that were employed in the experiments
were glucose, peptone and domestic sewage* These three are
different hinds of wastes ** glucose, a carbohydrate-waste,
peptone, a pnoteini ceous - waste and domestic sewage, an extre-
mely complex waste, The concentrations of the different substrates
in the experiments are shown in Table I,

d. Seeding Conditions?

r , Domestic .seed .was employed in the experiment in the con-

centration of 2*5 mg/lit * , S mg /lit., and 10.0 mg /lit. Inocu-
lation of the domestic seed: in the react! on- flasks ensured ■■
hetrogemaous microbial population to act on the different subs-
tracts. The ratios of seed to substrate (R) to study the effect
of bacteria! concentration on exertion were 1/8, 1/4 and

1/2. The seed was stored in the frigldaire through the entire
period of experimentation (3 months about}.

e. ... Jets of Experiments?

The experiments were conducted in Warburg- flas&3 of ■

125 ml capacity. The marietta sets of experiments are summarised
in Table t w ■

tmm i

3 B 9 P 3 or sxmswifrs

Substrate Initial deed Ratio of M-mbnr

substrata concentration Initial 0 f

concentration employed concentration S3 ts

(mg A) (rg/X) of Seed to

initial Substrate
Concentration

400

Glucose 240

2,5

1/160

6.0

1/80

10.0

1/40

2,6

■ ■ %/m

6 * 0 ' .

■ ■ 1/48

10.0

■ %/m

2*6

■ 1/48

5*0

■ 1 /M

10*0

1/12

2,6 1 / 1*0

400 6*0 . 1/80

; . 10 * 1 ® / ,; y " l/^;v

2.6 I / O ®

Peptone 240 5*0 1/48 '

10.0 1/M

2,5 1/48

120 ■ 5,0 1/M

10*0 1/12

2*6 %/m

Dome site Sewage 1®8 6*0 3/81 • 1

{as BOO) 10,0 1/18

Cl) BttfttMft water used im tfce meriataM t#' Ml m %p tbe
total w#liw»e mbatraie* aegt) to 60 »1 me prepared
neoevdlac to Standard Webbeda** 5 ® '

( 8 ) Temperature of Thermostatic bath « % , 1 # ' 0 *

CHAPTER y

experimental results

The data obtained free, the experimental studies
on the kinetics of BOD exertion and bacterial growth for
different substrate. (Glucose, Peptone and Domestic
Sewage) are presented in Appendix - B. The corresponding

The terms 'viable cell mass’ and ‘bacterial

concentration * have been synonymously used by the author
in all his works.

The symbol *1* marked @n the graphs is the
ratio of the initial concentration of the seed to the
initial substrate concentration and hence is a dimensionless
amber. This is a very signlfioaiffe parameter syggested
by the anther for river pollution studies and finds its
frequent nestles accordingly in the following chapters.

iMi

<:\y.

m m

'“"■*■:■ !■:

isssssis

|t 3 : - M. 3 M & # - - r t a/s

OSi

(<3U BOD £>#O6#£&$l0A/

a 5

-CHAPTER - VI

DISCUSSION AND ANALYSIS OP RESULTS

a. Theoretical Curves for BOD Progressions

The mathematical model vas programmed for computer opera-
tion and the numerical solutions were worked out for arbitrarily
chosen values of the constants (Kgj K x ) under assumed

initial conditions (X Q , S Q ) . These values are plotted in Figures
8 to 12 which demonstrate clearly the kinetic course of the
substrate depletion according to the proposed theory. Casual
observation of these graphs will at once reveal their significant
departure from the path of the Streeter-Phelps formulations. The
effect of variation in the values of the constants Kg, and

K on the geometry of the curves can be easily studied from
max

graphs 8, 9 and 10 respectively. It is interesting to note that

this effect is more pronounced for changes in the values of kinetic

constants K g and K ftax

changes from 1.5 to 2.5, (l.e. yield coefficient changing from

0*66 to 0.40), there is no much noticeable effect on the rate of

utilisation of substrate by the microorganisms. As expected from

the analysis of the equation of the model, the decreasing values

of K 0 result in a faster substrate- consumption and the effects
a

are just the opposite for stellar changes in K x •«>w Figures 11
and 12 present the graphic evidence for the effect of variation in
different initial conditions (X Q and S Q ) on substrate depletion .

than for the constant K^. Even though K x

TIME , HOURS

THEORETICAL CURVES FOR BOD PROGRESSION EFFECT OF VARIATION IN

TIME , HOURS (T)

FIG.9:-THE0RET!CAL CURVESs FOR BOD PROGRESSION [EFFECT OF

IME , HOURS

HOURS

OF VARIATION IN

I oo* — —

FIG.I2 :- THE0RETICAL CURVES FOR BOD PROGRESSION [EFFECT OF VARIATION

The significance of inclusion of a term representing bacterial
concentration in the suggested formulation for the description
of kinetics, is clearly borne out by the graphs in Figure 11*

Between the arbitrarily chosen range of 1 mg Ait* to 10 mg Ait.
for the initial bacterial mass, higher removals of substrate
are seen to be associated conceivably with the increased micro-
bial population. Figure 12 reassures the well known fact that
the rate of substrate utilization is proportional to concentra-
tion of the substrate itself, the concentration raised to some
power (the exact evaluation of which requires a rigorous mathe-
matical analysis of the equation (10) reported on page (24) .

Thus it is convincingly evident that the description of the
bio-oxidation kinetics Is adequately covered by the proposed
rational model and its graphical tracing offers ho problem in
the compilation of the graphs for various initial conditions,
in as much as the digital computer techniques are easily adoptable.

b. Theoretical Curves for Bacterial Growth:

The integrated version of the well known Monod * s equation
(reported on page vs ) provides a useful mathematical tool to
study the bacterial growth at different instants of time. The
influence and the inter-relationships of the various parameters
in the equation are clearly exhibited in figures 13 and 14*
Realisation of the fact that bacterial growth is proportional to
the substrate utilisation, makes it clear that the effect of variatic
In the values of the constants and Kg will remain the same

So « 4 <50

n"H)w ‘ssvw

for the kinetics of both substrate depletion and bacterial growth
under a given environment.

c# Agreement of the Observed Data with the Rational Model;

Ihe experimental studies carried out on the biological
oxidation of substrates like Glucose, Peptone and domestic sewage
clearly bear evidence to the theory proposed# The observations
indicate that the rate of substrate utilization by the microbial
flora depends not only on the instantaneous substrate concentra-
tion as a monomole cular kinetics would suggest, but also on the
instantaneous concentration of the micro-organisms. It is a
well known fact that the ultimate foundations underlying the bio-
cheml cal oxygen demand of waste materials are the enzyme catalysed
processes involved in the growth and multiplication of the organisms
acting on these materials. In view of the significant role played
by bacteria in the removal of substrate, caution should be exercised
in the selection of proper type of seed and its amount while per-
forming the BOD tests by the standard delutioa technique.

fable II shows the effect of the different seed to substrate
ratio on the BoD progression of substrates viz.. Glucose, Peptone
and domestic Sewage. It can be noted that for a given initial
seed, the rate of substrate depletion increases with the correspond-
ing increase in the substrate concentration. The choice of domestic
seed in the experiment was made in particular because of its common
presence in the polluted rivers, for which the rational model was
proposed. This has also given the opportunity to study in
details the ability of the complex microbial flora present in

th« seed In consuming simple (glucose) as well as cop- pie*
(swage) substrate. Prop Sable 21 it is alto observed that
the rate of Glucose utilisation remains more or the less the
same as that of savage* This finding goes to substantiate
the conclusions arrived at by investigators like Garret,
■lawyer andGGaudy that the kinetics of the removal of com pie*
substrates follows the same relationship that are applicable
to the? utilisation of simple substrates.

TABU 21

Substrate

it lu cose

Initial concen-

Seed to subs-

BGb escorted

tration of subs-

trate ratio.

end of 48 h:

trate (on BOD

$ ge of the

in Kg A *

rail cal 3Gb

1/160

418

1/80

1/40

i/m

262

1/24

1/48

131

1/24

1/12

Peptone

460

872

138

1/160

1/60

1/40

1/86

1/48

1/84

1/48

1/24

1/12

1/72

1/36

1/18

borne stlc sewage 168

5 2

Figure 15 brings out in essence the striking resemblence
of the observed data to the postulated model* The graphs shown
in the figure correspond to the kinetics of substrate utilization
and bacterial growth for the three substrates, acted upoit by
Identical bacterial mass. Only 3 sets of readings are recorded in
the graphs corresponding to the initial substrate concentrations of
400 mg/lit* for Glucose, 400 mg /lit, for Peptone and 182 mg/lit,
for domestic sewage. The initial bacterial concentration adopted
was 2*5 mg/lit. The values of the constants (K x , K g and in

the theoretical calculations for BOD exersion and Bacterial growth
were taken from the graphical plot made in fig. 19 for the
experimental observations. It is very clear that the observations
for BOD exertion fit closely to the proposed model and those for
bacterial growth, to the Monod equation.

Referring back to Fig. 1 to 7, showing the experimental
observations, the presence of a lag period can be noticed, especially
at lower concentration of the seed, in the early portion of the
bacterial growth curves after which they rise concave upward* The
ratinnal model presented by the author does not account for this
type of kinetics with a lag period. The presence of the lag period
is attributable to the inability of the reduced microbial population
in the initial stages to consume the substrate,
d. Comparison with the Streeter- Phelps Formulation j

With the distinct! d features of the proposed model well
recognized, one can strike a comparison between the two formula*
tions — the rational and the first order, primarily with a
view to pinpoint the inadequacy of the later equation in des-

crlblng the substrate depletion kinetics. First of all., the
application of the monomolecular equation is empirical in
nature and the ultimate demand is entirely theoretical*
neither of the two constants* (K* and L) that appear in the
equation can be determined directly, fhey call for an extensive
application of *the curve fitting techniques such as those suggested

' *3j>

by Theriault , Thomas 3 and others* I can not be determined
experimentally because it is BOD at infinite time. Thus K' and L
as ordinarily calculated serve only as statistical constants to
shape the BOD curve instead of acting as physical and biological
parameters*

The application of the proposed model, on the other hand, is
rational because none of the constants in the formulation is hypo-
thetical, % stands for the degree of conversion of the substrate

into the protoplasmic mass while XL and K go to describe ade-

o max

quately the bacterial kinetics* The initial conditions are well

recorded in the presence of the parameters XL and 8 * The forou-

u o

lation attempts fairly satisfactorily in the correlation of the
BOD removal kinetics with the an z ymes and growth kinetics in
bacteria.

Figures 16 , 17 and 18 clearly record the comparison of the
two formulations. In the early stages of BOD exertion the first
order equation stipulates a higher percentage of BOD consumed

* The constants are the rate constant ft*, and the ultimate oxygen
demand I, which appear in the Streeter- Phelps equation,

Y ■ h (1-e ~ Klt ) where, Y - BOD exerted in time t*

TIME , HOURS

*NouttifJLN30NQO jiMUttirns

FIG.ie COMPARISON OF STREETER -PHELPS AND RATIONAL * MODEL

TIME y HOURS

■in/$fr < AtQ/J.t/y±N33MQ 0

COMPARtSON OF STREETS R r PHELPS MD RRT/ONRL &mODEl

TIME* HOURS

COMPARISON OF STREETER - PHELPS AND RATIONAL MODEL

5 ?

than what is postulated by the rational equation, The values of
the: constants Kg, K y and K ^y war® chosoa arbitrarily and the
graphs, traced for the computed values in accordance with the
mathematical model* The fitting of the Streeter- Phelps curves
was done by least-squares method* It is noticed that while
these curves tend to remain asymptotic to the time-axis with
increasing time* the rational ones droop down considerably*

These graphs thus provide sufficient Insight into the descrip-
tions of kinetics of substrate depletion and one can obviously
conclude that the first order equation is highly unjustified
in the light of the existing knowledge on the kinetics of BOD
reactions*

e* Determinati on of the Const ants i

A major task in the verification of the presented model
with the observed data lies in the evaluation of the constants
Kg and i^ ax * The value of %, as far as theory goes,
should remain constant for the same substrate and seed condi-
tions. The reported values in Table I? however contradict
this and the fluctuations in the values of % are mainly
attributable to the change* in behaviour of the enviromtent
characterised by the hetrogeneous microbial population, varies
from 1*1 to 3*7 for Glucose, from 1*7 to 3*2 for f*eptcne and
from 2*4 to 3*7 for domestic sewage*

Tmm m

VALUES OF FROM OBSf3*V?50 DATA

Substrata

Initial concentration
of substrate (mg/1)

Seed to subs* Values

trate ratio (R) of K Y

Glucose

Peptone

1/160

2.3

400

1/80

1,6

1/40

3.7

1/96

1.8

240

1/48

2.6

1/24

2,7

1/48

1,1

120

1/24

1.7

1/12

2,2

1/160

1.7

400

1/80

3.1

1/40

4.0

1/96

3.2

240

1/48

2,8

1/24

2,7

1/48

1,8

120

1/24

1,7

1/22

2.4

domestic Savage

182

1/72

1/36

1/IP

2.4

2.9

3.7

The evaluation of the Kinetic constants and Is

facilitated by the linear plot of the Monod»s equation as
suggested by Llneveaver and Bert, tj * ®

Kg 4 S

Taking reciprocals on both sides of the above equation.

l =

k 3

a.#

It can be seen that by plotting 1/u vs* 1/S, a straight line
of slope %/^ a3[ and intercept 1/K^^ should result, if the
Vichaelis equation holds* * Figure 19 shows such a plot for
the constants, K i and K^ ax *

TAILS I?

VAI.TTJK OF K, hNu K. FROM TKS 0BS3RV3) 3A7A

substrate

% (mg Ait.)

■Vax to' 1 )

Glucose

660

0 §22

Peptone

636

0.25

Sewage

418

0.23

The major revelations of this table are t© the values

of iC Increases as the substrate is »o re and more complex
TMUC

(2) the values of K s decreases with increasing complexity
of the substrate, which should be expected in the light of
arguments under the subheading (a) in this chapter. It was
assumed that the endogenous respiration will be negligible
in the Warburg - Flasks for the time of duration of the
experiment (48 hours)* To account for the endogenous uptake

fix/s Mr rc*

Mentioned equation vu Modified by Tend, and Horton ae
follows 1

- l. s „ K a / t v 4 . i

^ 3 ' ^ ^nax * fV

where , ^i 0 is the respiration rate
with no added substrate.

Careful observation shows that there need not be any
conflict over the discrepancies in the value of Kg and J^, ax
reported by early investigators. Tench and Morton have
obtained the values of Kg for Glucose and Peptone as 133 and
29® mgA.it * respectively. It should be borne in mind that
they used in their experiments the activated - sludge-seed
which harbours some organisms quite uncommonly noticed in the
domestic.^seed, (e.g. Sphaerotilus natans), Since the author
has adopted small Initial concentrations of the micro-organisms
as contrasted to the mass culture work published by many research-
workers , there is bound to fee some inconsistency In the reported
values of X and Garret and Sawyer 10 have indicated through

their experiments, the values of as 0, 18/hr and 0.21/hr
respectively for glucose and peptone at 2Q°C, They postulated
that only two phases — a log phase and a transition to the
stationary phase — * war® of practical importance in the defini-
tion of the reaction, kinetics of the aerobic biological processes*
They objected Monad * s theory on the basis of their experimental
results and suggested that ’the equation denies the existence of .
a constant rate of growth above critical concentrations of food,
although this Is the most frequently observed phenomenon related
to the growth of bacteria’ # However, experimental data from

iilson's experiments were found to fit dose to the ff©nod*8
equation rather th«a to th«« tv*»jfaag« formulation* Savor al eyelet
of experiment* or® needed supporting the validity of the tvo
theorl*», Th# author** own experiments have shown that tossed**
equation provide* a striking fit to the observed data*

i) hraotiool Applications;

he significance of the rational twdd lias in its direct
application to the field condition** It can b# very easily adopted
by the agencies concerned in the vater* polluti otw&bat eeent program-
net « Its amenability for solution by modern digital eonputer
techniques should provide adequate attraction especially to those
with whom lies the responsibility of the data collection in river*
pollution studies* lust the different sets of BvJ3*graphs said to
ho compiled and thereafter it regains to match the observed pro*
files of pollution with the theoretical graphs*

CHAPTER VII

CONCLUSIONS

A deeper understanding of the kinetics involved in the
substrate depletion is essential for adoption of methods to
predict the exact degree of organic pollution in streams# The
author has drawn the following conclusion® based on his
experimental works,

1* The rate of progression of BOD in streams is found to
he proportional not only to the concentration of the remaining
B.O.D. at that instant but also to the bacterial concentration.

The experimental observations validate the proposed theory for
the BOD kinetics based on a realistic and rational approach.

2. The bacterial growth observed in the experiments
follows the curve typified by Monod ' s equation. The two phase
formulations of Garret and Sawyer do not hold good in particular
for the author * s observations . No existence of the ‘plateau’
reported by Busch et al was found in the geometry of the BOD
curves.

3. The rational model has several advantages over the first
order formulation. First of all, its application for practical
problems is quite simple and direct. The extensive use of the
curve fitting techniques required for the first-order-reaction -
kinetics is done away with. All that one has to do is to match
the experimentally observed curves with the theoretical graphs

LIST OF REFERENCES

1* Streeter, H.W., and Phelps, E.B., "A Study of
Pollution and Natural Purification of the Ohio
River. * U.S. Public Health Bulletin No. 146 (1925).

2* Oxford, E.E. , and Ingram ¥.T. , "Be oxygenation of

Sewage." Sewage and Industrial Wastes, 25, 419, (1953).

3. Thomas, H.A., Jr., "Analysis of BOD Curves.” Sewage
Works Journal, 12 , 3 , 504 (1940).

4. Fair, C.M. , "Log*Difference Method of Estimating the
Constants of the First Stage BOD Curve.” Journal of
Water Pollution Control Federation, 3g ^ 673 , ( 1931)

5. Monod, J. , "The Growth of Bacterial Cultures." Annual
Review of Microbiology, 3, 371 (1949).

6. Busch, A.W., "BOD Progression in Soluble Substrates.”

Sewage and Industrial Wastes, 30, 11, 1336 (1958).

7. Gaudy, A.F., "Factors affecting BOD Plateau.” Journal

of Water Pollution and Control Federation, 37, 4, (1965).

8. Busch, A.W. , "A Short Term BOD Test.” Journal of Water
Pollution Control Federation, 36, 3 (1963).

9. Butterfield, C.T., "Experimental Studies of Natural Purifi-
cation in Polluted Waters. "Pub .Health Report ,44,2865 (1929).

10. Hinshelwood , C.N., "The Chemical Activities of the
Bacterial Cell." Clarenden Press, Oxford. (1946).

11. Herbert D., ”A Theoretical Analysis of Continuous
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Downing, A.L., and Wheatland A.B., "Paper presented to Mid.
Branch of Inst, of Chem. Sngg. (i960).

Garret, T.M. , and Sawyer C.N., "Kinetics of Removal of
Soluble BOD by Activated Sludge." Proceeding of Seventh
Industrial Waste Conference, pp. 51-77, Purdue University
(1952) .

Eckenf elder, W.V. , and O'Connor, D.J., "Biological Waste
Treatment." Pargamon Press (1959),

Wilson, I.S., "Concentration Effects in the Biological
Oxidation of Trade Wastes." Proceedings of the First
International Conference on Water Pollution Control
Research, September 1962, 2, 27, Pergamon Press (1964), .1

Young, J.C, and Clark, J.W., "Second Order Equation for

BOD," Journal of Sanitary Engineering, Division, Proceedings j
American Society of Civil Engineers, ftl, SAI, 43 (1965). j

ReVelle, C.S., Lynn, W.R. and Rivera, M.A., “Bio- Oxidation
Kinetics and a Second Order Equation Describing the BOD
Reaction,” Journal Water Pollution Control Federation,

22, 1679 (1965).

Thayer, P., and Krutchkoff, R.G,, "Stochastic Model for
BOD and DO in Streams* Journal of Scientific Engineering
ASCE, Yol. 93 NO, SA3, g9, (1967).

Gram.A.L., "Reaction Kinetics of Aerobic Biological Processes^
Report No. 2, Sanitary Engineering Research Laboratory,
University of California, Berkeley (1956).

Stewart, M.A., and Ludwig H.F., "Theory of MAS Waste-Water
Treatment Process." Water and Sewage works, 109, Nos. 2 and

21* MeGahey, P*H., "Discussion on I.S. ¥ilson*s Experiments*”
Proceedings of the Internatloneal Conference held in
London, Edited by W.W. Bcfcenf elder , Pergamon Press,
pp* 44 to 46, (1964)*

22* Keshavan , K. , 3ehn, V*C* , and Asses, W,F. , "ELneti cs of
Aerobic Removal of Grgahic Wastes*” Journal of Sanitary
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23. Keshavan, X., "Discussion on Second Order Equation for BOD,”
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American Society of Civil Engineers, SAl , 91 (1965),

24. Reiner , A. J *, "Behaviour of Enzyme Systems,” Bumgess Co, (1959)

25. McKinney, R.S., "Mathematics of Complete Mixing Activated
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American Society of Civil Engineers, Vol. 88, Paper 3133
(1962).

26. Helmtrs, E.N., Frame, J.D., Greenberg, A.E, , and Sawyer,

C.H., "Nutritional Requirements in the Biological Stabili-
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1077 ( 1962) »

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Methods, xsth Edition.

' an ^ ® ar * t ®*» Journal of American Che®ical

Bnginaarinc Society, 06, 658 C1S34).

Tonidi H.B* and Horton A.Y, «?h® Application of Snsyse
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Theriault, S.J., "The Oxygen Baaand of Polluted waters."
Rub. Health Bulletin Ho. 173 (192?) *

GLOSSY OP SBSQLS

K • fh® rat* constant ia tho first- or dsr BOB o^atloa
*te * «iete«Hs*%fi«n Constant

*i»a:r K Growth %t« of the? Baatorial Calls

« Saturation Constant

*Sc E litt<li P ro ««l Constant of th# Yiald Cosffldsat
I* *■ Glhlisate OKYgsn in the flrst—or&ar BOB

aquation

B * Substrata eoaosntratloo

I • Tln«

£ * Growth rata of baetarlal calls

X * Baatsrial mmmmmUm or VlahiLo mm *xpr*»s*0

in conoantraticn units.

I //ABLE. CELL CONCENTRATION, MC./L’d

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TABI5 OF OBSERVATIONS

mmm & g

TABtE FOR FLASK OOKSTAHTS

Fla*k Constant, k * Zg * Y f j l

ihere, » volume of gas spans in the
vassal (including the completing and
manometer tubas down to the reference-
mark)

l*o - Absolute Temperature (Kelvin) « 273OK
T * Absolute temperature of the thermostatic
bath,

» Volume of fluid in the vessel ,

* Solubility of the evolved gas (COg)
in the liquid in the vessel * 0,0361
(expressed as ul of gas at N.T.P.
dissolved in l ul liquid when in
equilibrium with a partial pressure
of the gas equal to P 0 )

» Formal pressure in s m of manometrle
fluid, for Brodies solution, * 10000 saw.

•FLASK WJMBSP FLASK CMSTAJtt

k in pl/m®

6.39

* Flasks of 126ml volume

6.76

6.88 ■

v r oS0 " 1

JKhk

f * 26°C

4 (Thermo

6.63

barometer)

integration op the differential equations ( lo ) and (fa.)

BOD Equation:

dS = 5

K s + S

where,

where, X = X Q + (S Q - S)

Substituting the value of X in the above equation, and rearranging,

~ ds ( K S + s >

<x 0 + i^ - (S 0 -S) ) s

h K »a* dt

Now,

Kq + S

(x + -= — (S -S) )S
o o ' '

V -XJ (3o-S)

i*e.

At S * S Q ,
At S * 0,

i.e. A(S) + B <X Q +

S -S) - K™ + S

Ky 0 3

A <S Q ) + B (X 0 ) * K s + S 0
B (X 0 + (S q /K x ) * K g

B = ( 2 )

V ( V®*>

Substituting this value in (2) and solving for A,

A * l/* 0 (K s * 3 J - ^...Q

APPKRDIX * D-I (Continued)
*» *<

X .4

(Sq/ICjj)

Substituting the values of a and B in (l), rearranging, end
integrating both sides,

1 +

£ (y 4 -i* s >

X lA o % s o'j

wIh 8 m

(3C o^ V 5 >

v O

Kjr Kfeax «

*f> flRHKpKF

i.e.

T «JL» <5

V s o

* log (Xq * (l/X^) 3^*8 ) ♦ log S

J 3 f

log (x 0 * (lAj) vs

fx *i«x *

i*e<

■3

* (1/£ 5C )S 0

X ♦ (1/K X ) IJs

log (Sj/S) f

Kg log

i * a/%

**»* t

i #01

(x„ * a/%) s c -s

4 If

1 %

*X ^x *

[£nliiglLicg

i.e. (| * O/Kjj X 0 ) S 0 .S )

<-VS)

liMr

„ o

X- /
o

% ^x t

4 V (1/K x^ 8 o

Cl./--.,)

X * (l/%)

CS/3)

V (V%)S 0

7CV

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